Coordinated Tracking Control of Multiple Laboratory Helicopters: Centralized and De-Centralized Design Approaches

Coordinated Tracking Control of Multiple Laboratory Helicopters: Centralized and De-Centralized Design Approaches Hugh H. T. Liu University of Toronto, Toronto, Ontario, M3H 5T6, Canada Sebastian Nowotny Universität Stuttgart, Stuttgart, Germany This paper presents coordinated control among several laboratory three-degree-of-freedom (3-DOF) helicopters in their serial tracking maneuver. Both centralized and de-centralized design approaches are investigated using linear quadratic techniques. The experimental hardware-in-the-loop results are presented and their comparative analysis is provided. I. Introduction One can find the coordinated control of multiple objects in many industry applications, such as the multiaxis motion control in manufacturing process, the cooperative control of robot manipulators, and formation flight in aerospace. Among many design challenges, trajectory tracking and motion synchronization is the topic of our interest. Our testbed is a set of three (3) laboratory helicopters (robots) that can perform the elevation, pitching and travel motions. Therefore, they are called the 3 degree-of-freedom (3-DOF) helicopters. Control on a single laboratory facility that is the same or similar to ours has been investigated before, such as in references. 3 In this paper, we investigate the coordinated control strategies for multiple 3- DOF helicopters in a serial tracking and synchronization configuration. We use the linear quadratic tracking technique to design the controllers. Both centralized and de-centralized design approached are investigated. Furthermore, the designed controllers are implemented. Numerical simulations and hardware-in-the-loop experiments have been conducted. Their results and comparative analysis is provided. The rest of this paper is organized as follows. Section 2 describes the helicopter system and the model development. In section 3, the linear quadratic tracking technique is briefly introduced. Section 4 presents the de-centralized and centralized coordinate control strategies, followed by the simulation and experimental results in Section 5. Finally, Section 6 provides the concluding remarks. II. Laboratory Helicopter System and Model The laboratory of the Flight Systems and Control Group at the Institute for Aerospace Studies of the University of Toronto (UTIAS) features four 3-DOF Helicopters from Quanser, 4 as shown in Figure (a). For this study, three of them were connected to be controlled from one computer to enable formation flight. The 3-DOF Helicopter resembles a tandem-rotor helicopter like the Boeing CH-47 Chinook. It is linked to a base by three rotating rods. The vertical pole enables the helicopter to turn around the base. This angle is defined as travel angle λ. The second degree of freedom is the rotation of a rod around a pivot point on top of the vertical pole enabling up- and downward movement around the elevation axis ε. The third motion is the pitch angle θ. A bar with a motor/propeller combination at each end can swivel about its center perpendicular to the elevation axis. The propellers create a force normal to the elevation-pitch plane. If the same voltage is applied to both motors, the helicopter rotates around the elevation axis. By applying Assistant Professor, Institute for Aerospace Studies (UTIAS), 4925 Dufferin Street, AIAA Member Exchange Research Intern to the UTIAS, Institut für Flugmechanik und Flugregelung of 8

a differential voltage, it turns around the pitch axis. The travel angle is controlled by the pitch angle. Note that a positive pitch angle will result in a negative travel rate. (a) UTIAS.FSC Helicopter (b) Helicopter Model Figure. Laboratory Helicopter System and Model To derive the differential equations for a model, the 3-DOF Helicopter was discretized. Three pointmasses represent the motor/propeller combinations and the counterweight as shown in figure (b). The connecting rods were assumed massless and rigid. No friction at the pivot points was taken into account nor any drag of the moving helicopter. The forces generated by the propellers were assumed linear to the input voltage of the driving motors as F = K f V. () The three differential equations describing the model were calculated using the method of virtual displacements. 5 They are Elevation Axis ε: ε = l a cos(θ) K f J ε (V f + V b ) g l a m eff cos(ε) J ε (2) Pitch Axis θ: Travel Axis λ: θ = K f l h J θ (V f V b ) (3) λ = sin(θ) Kf (V f + V b ) l a J λ cos(ε) (4) In order to use a state space approach, the model was linearized at the following positions: ε =, θ = and λ =. Elevation Axis ε: Pitch Axis θ: Travel Axis λ: ε = l a K f Jε ( V f + V b ) (5) θ = K f l h J θ ( V f V b ) (6) λ = g l a m eff J λ θ (7) 2 of 8

The state-space equation then becomes ε θ λ ε θ λ = g la meff J λ ẋ = A x(t) + B u(t) (8) ε θ λ ε θ λ + l a K f J ε K f l h J θ l a K f J ε Kf lh J θ The values for the 3-DOF Helicopter are given in table and table 2 describes the linear system. Description Symbol Value Unit Gravitational acceleration g = 9.8 m/s 2 Mass of counterweight m w =.87 kg Mass of helicopter m h =.44 kg Arm length from elevation axis to helicopter body l a =.65 m Distance from pitch axis to either motor l h =.36 m Distance from elevation axis to counterweight l w =.47 m Motor force constant K f =.625 N/V Table. Properties of the 3-DOF Helicopter V f V b. (9) ε θ λ ε θ λ V f V b ε θ λ ε.398.398 θ 2.44-2.44 λ -.526 Table 2. Linear system of the 3-DOF Helicopter III. LQ Tracker The approach presented here is based on a linear quadratic regulator (LQR) which is converted into a tracker. 6,7 It is important to note that for perfect tracking, the number of command signals to track has to be the same or less than the number of control inputs in vector u. The original system from equation 8 written here as ẋ = A x (t) + B u(t) () is augmented with the tracking error e(t), which is calculated by e = r z. () r(t) is the reference input and z(t) is the performance output, that tracks the reference input. The the performance output z(t) is defined as x z = H. (2) e 3 of 8

The augmented system then becomes ẋ A = e H The state feedback is chosen in the form of u = x e + K K e B x e u + I 2 r (3) (4) where the feedback of integral term is introduced to improve the tracking performance. H is defined as H = (5) This yields In detail ε θ λ ε θ λ e ε e λ = + ẋ = Ax + Bu + Er z = Hx (6) u = Kx. glameff J λ l ak f J ε K fl h J θ l ak f J ε Kflh J θ V f V b + The structure of this tracking controller is presented in figure 2. Assuming that the reference input r(t) is a step command of magnitude r, the resulting gain matrix will stabilize the system for any reference command r(t). To calculate the optimal K, an optimal control problem is formulated, by transforming the system in 6 into a deviation system. The equations satisfy the deviation system ε θ λ ε θ λ ζ γ. (7) x(t) = x(t) x (8) z(t) = z(t) z = H x(t) (9) ũ(t) = u(t) ū = Kx(t) ( K x) = K x(t) (2) ẽ(t) = e(t) ē (2) x = A x + Bũ (22) z = H x (23) ũ = K x. (24) 4 of 8

z = Hx H - r + e int(e) + u x /s K e Plant - K Figure 2. Tracking control structure for one helicopter With equation the deviation error is The performance index to minimize is ẽ = z. (25) with and J = Q = diag ( x T Q x + ũ T Rũ ) dt (26) R = diag q q 2 q 8 (27) r r 2. (28) The performance index which was used to calculate the gain matrix, makes the deviation system small, and thus ẽ. But since e = ẽ + ē, it has to be taken into account, to make ē small as well to ensure that e is minimized. Integrators in the feed forward loops were added to address this problem. Since the steady-state error of the state x is not equal to zero, it is calculated to show, that it can be neglected because of its small order. Assuming a step input of r, becomes Solving for the steady-state vector x yields lim t ẋ = lim (A BK)x + Er (29) t = (A BK) x + Er. (3) x = (A BK) Er. (3) With the values of table 2 for the system matrices, E = (32) 5 of 8

and the gain matrix given in table 3, that will be used for the simulation in chapter?? as well as T r =, (33) the steady-state value x is equal to and the steady-state error ē is x = ē =.9924 4.2739 (34). (35) IV. Coordinated Control for Serial Synchronization Configuration Once a controller for the single 3-DOF Helicopter has been found, the setup is expanded to multiple helicopters. A serial setup of tracking configuration among these helicopters is adopted. The first helicopter tracks a command, e.g. a given trajectory or joystick input (, ), while the second helicopter tracks the performance output of the first (, ), the third the performance output of the second (, ) and so on. The controllers of all agents use the same gain matrix, which is designed for a reference input of a unit step with magnitude r. Variables of the first system are denoted with the subscript of, those of the second system are indicated with the subscript of 2 et cetera. In this paper, three helicopters are used. The tracking errors for each helicopter are given as e = r z e 2 = z z 2 (36) e 3 = z 2 z. The augmented state-space system for three helicopters is expressed as follows ẋ A x ẋ 2 A 2 x 2 ẋ 3 e = A 3 H x 3 e e 2 H H e2 e 3 H H e3 + B B 2 B 3 u u 2 u 3 + I r (37) 6 of 8

A. De-Centralized Design Approach The decentralized controller has the same connections between the helicopters as the controller for the serial setup, but the gain matrix is calculated for the complete group of helicopters at once. The augmented system remains the same shown in 36 and 37, but the state-feedback now becomes u u 2 u 3 = = K K e K 22 K e22 K 33 K e33 K K e x e x x 2 x 3 e e2 e3 (38) B. Centralized Design Approach For the centralized control, the tracking errors and the state-space system are again the same as shown in ( 36 and 37. The cost function to be optimized is J = x T Qx + u T Ru ) dt. But since the off-diagonal terms of the gain matrix are not forced to zero as in the previous section A, the state-feedback now becomes u u 2 u 3 = = K K 2 K 3 K e K e2 K e3 K 2 K 22 K 23 K e2 K e22 K e23 K 3 K 32 K 33 K e3 K e32 K e33 K K e x x 2 x 3 e e2 e3 x. (39) e As can be seen in figure 3, the gain matrices K and K e and the output matrix H can now easily be implemented in Simulink. This control structure complies to all three control strategies and therefore only K and K e have to be changed to switch the controller. z H z 2 H z 3 H r - + - + - + e /s int(e) K e + - u Plant x x 2 x 3 K Figure 3. Tracking control structure for three helicopters 7 of 8

V. Simulations, Experiments and Analysis To evaluate the performance of the group flying in formation, for each helicopter the L 2 Norm of the errors between its command and the state was calculated as well as the sum of all errors as shown in equations 4. This was done separately for elevation and travel. e ε = e ε2 = e ε23 = e λ = (4) e λ2 = e λ23 = E ε E λ = e ε + e ε2 + e ε23 = e λ + e λ2 + e λ23 The performance of the controller is evaluated in several steps. The first examination is done with the nonlinear single helicopter model. In the second step, the step-input responses of all three 3-DOF helicopters are observed using both the de-centralized and centralized approaches. When the simulation results are satisfactory, the controllers are implemented into the hardware-in-the-loop system. Experimental results are then presented and analyzed. A. Sniggle Helicopter: Nonlinear Simulations For each single helicopter, a set of Q and R matrices was found that yielded a gain matrix with fast rise and settling time. With Q = diag5 5 4 3 and R = diag the gains shown in table 3 were obtained. ε θ λ ε θ λ ζ γ V f 7.763 7.3546 9.5567 4.4593.8742 5.9498 3.873 2.236 V b 7.763 7.3546 9.5567 4.4593.8742 5.9498 3.873 2.236 Table 3. Gains for the single helicopter The response of the nonlinear model to the elevation step input is shown in Figure 4(a). The voltage for hovering in a horizontal position is not zero for the nonlinear model. The most noticeable difference is the drop in elevation due to the commanded pitch angle for the travel step. But a deviation of.2 deg is hardly noticeable. Table 4 shows the tracking errors for both models. The performance of the controller with the nonlinear model comes very close to the ideal performance. model controller step e ε e λ state-space single ε 774.6. λ. 36. nonlinear single ε 776.. λ 6. 362.2 HIL single ε 83.9 9.2 λ 67.7 3733.9 Table 4. Tracking errors for a single helicopter B. Three Helicopters: Responses to Step Input With the good results for the single helicopter tracking controller, the design was expanded to three helicopters. Both the de-centralized control and centralized control are tested under the serial tracking and 8 of 8

5..2 5 5 ε.5.5 λ θ...2 ε 8 6 4 2 λ θ 5 5 5 2. 5 5 2.3 5 5 2 2 5 5 2 3 25 2 5 5 ε V.9.8.7.6 V f V b 5 λ V.9.8.7.6.5 V f V b 5 5 5 2.5 5 5 2 5 5 2.4 5 5 2 (a) response to elevation step input (b) response to travel step input Figure 4. Single Nonlinear Helicopter Model Control synchronization configuration. The control strategies are both based on LQR. This means, that the only design parameter that can be varied now are the weighting matrices Q and R. As a starting point, the weighting matrix for one helicopter was extended to the three model lineup, giving equal matrices for the three methods. The resulting gain matrices are presented in Table 5 and 6 for the decentralized control and the centralized control respectively. The simulation results with the linear state-space model are presented in Figures 5(a), 5(b), 6(a) and 6(b). The characteristic of the decentralised controller in figure 5(a) shows a slower response for the first helicopter, but better results for the following helicopters. It can be seen in table 9 a which shows the tracking errors for each control method in the initial design step, that the overall tracking performance of the decentralised approach is better. The centralised controller shows even better results, but also the slowest response for the first helicopter as can be seen in figure 5(b). Similar behavior can be observed in the travel step response. The decentralized system in 6(a) has a very slow rise time for the second and third helicopter. Therefore, the tracking performance is worse than for the serial setup. This is an issue which has to be addressed in the next design step. Figure 6(b) for the centralized controller shows the potential of this approach. The first helicopter tracks the command almost as good as a single helicopter, and the other two agents follow very close. The figures for the nonlinear models show the similar pattern. The differences with the linear state-space model are very small. Detailed results are omitted from this paper. To achieve the maximum performance for each control strategy, the weighting matrix Q was tuned for each controller individually. The resulting gain matrices can be seen in table 7 and 8. All tracking errors for the tests with improved gain matrices are listed in table. For completeness, the values for a so-called serial setup are also included in this table, where each helicopter control gains are chosen the same as the single control ones. The response to both, elevation and travel step, has improved for the decentralized control, as can be seen in figures 7(a) and 8(a) for the state-space system. The third helicopter still needs a long time to reach the desired travel command, but the tracking errors are now smaller than those of the serial setup. The behavior of the centralized controller has improved as well. The step response of the state-space system are shown in figure 7(b) and 8(b). The graphs of all three helicopters are very close together now. When running hardware-in-the-loop tests with this setup, the differences between the helicopters are not visible for a spectator. a Note some simulation and experimental results are omitted in this paper, since the presented results are representative. Detailed tracking performance values, however, are included in the table for completeness and for comparison purposes. 9 of 8

θ θ λ ζ γ V f 8.8888 7.792 2.5582 3.8793.824 7.329 3.729 2.483 V b 8.8888 7.792 2.5582 3.8793.824 7.329 3.729 2.483 ζ 2 γ 2 V f V b 9.3356 7.7889 2.994 4.3672.7876 6.6498 3.8638.8882 9.3356 7.7889 2.994 4.3672.7876 6.6498 3.8638.8882 θ3 λ3 ζ 3 γ 3 V f V b 8.2846 7.4344 9.9942 4.983.823 5.923 5.65 2.535 8.2846 7.4344 9.9942 4.983.823 5.923 5.65 2.535 Table 5. Gains for the initial design step, decentralized control θ θ λ ζ γ V f 8.668 7.5342.549 4.6947.8936 7.398 3.3738.9479 V b 8.668 7.5342.549 4.6947.8936 7.398 3.3738.9479.7.255.832.276.234.6282.55.8669.7.255.832.276.234.6282.55.8669.2449.67.4627.789.75.4875.674.674.2449.67.4627.789.75.4875.674.674 θ2 λ2 ζ 2 γ 2 V f.7.255.832.276.234.6282.8723.8 V b.7.255.832.276.234.6282.8723.8 8.369 7.4672.6922 4.658.886 6.8323 3.397.755 8.369 7.4672.6922 4.658.886 6.8323 3.397.755.2565.2825 2.2939.3549.39 2.57.55.8669.2565.2825 2.2939.3549.39 2.57.55.8669 θ3 λ3 ζ 3 γ 3 V f.2449.67.4627.789.75.4875.334.929 V b.2449.67.4627.789.75.4875.334.929.2565.2825 2.2939.3549.39 2.57.8723.8.2565.2825 2.2939.3549.39 2.57.8723.8 7.353 7.257 8.86 4.3398.8627 5.24 3.3738.9479 7.353 7.257 8.86 4.3398.8627 5.24 3.3738.9479 Table 6. Gains for the initial design step, centralized control of 8

2. 2..5.5.5.5 2 2 3 3. 2 3. 2 2 3 3. 2 3. 2.5 2.5.5.5 2 3.6. 2 3. 2 3.6. 2 3..4 V f V b.5 θ.4 V f V b.5 θ V.2 V.2.5.5.2 2 3. 2 3.2 2 3. 2 3 (a) decentralized coordinated control (b) centralized coordinated control Figure 5. Three Helicopters Coordinated Control: Responses to the Elevation Step Input...5.5 5 5.5.5. 2 3. 2 3. 2 3. 2 3.5.5 5 5.5.5. 2 3.4 2 3. 2 3.4 2 3.2 V f V b.2 V f V b V.2 θ V.2 θ.4 2 3 2 2 3.4 2 3 2 2 3 (a) decentralized coordinated control (b) centralized coordinated control Figure 6. Three Helicopters Coordinated Control: Responses to the Travel Step Input of 8

θ θ λ ζ γ V f 2.362 7.2 5.7326 4.3982.6838 6.2 5.6 2.3266 V b 2.362 7.2 5.7326 4.3982.6838 6.2 5.6 2.3266 ζ 2 γ 2 V f V b 3.443 7.3546 7.7266 5.746.748 8.6727.337 4.23 3.443 7.3546 7.7266 5.746.748 8.6727.337 4.23 θ2 λ2 ζ 2 γ 2 V f V b 9.624 6.7284 2.7372 5.5227.7423 7.296 8.826 3.989 9.624 6.7284 2.7372 5.5227.7423 7.296 8.826 3.989 Table 7. Gains for the final design step, decentralized control θ θ λ ζ γ V f 3.57 6.5758 2.3267 9.59.786 6.626 5.5656.789 V b 3.57 6.5758 2.3267 9.59.786 6.626 5.5656.789 4.93.4783 4.74.5462.55 3.3337 2.3395.776 4.93.4783 4.74.5462.55 3.3337 2.3395.776.339.87.823.95.29.93.8843.943.339.87.823.95.29.93.8843.943 θ2 λ2 ζ 2 γ 2 V f 2.29.4798 4.583.5462.55 3.3577 4.573 2.7827 V b 2.29.4798 4.583.5462.55 3.3577 4.573 2.7827 2.795 6.78 3.6898 5.52.8 7.66 8.2935 3.25 2.795 6.78 3.6898 5.52.8 7.66 8.2935 3.25.7345.446 3.647.432.5 3.53 3.2.49.7345.446 3.647.432.5 3.53 3.2.49 θ2 λ2 ζ 2 γ 2 V f.6242.835.2827.95.29.2224.86.5732 V b.6242.835.2827.95.29.2224.86.5732.733.443 3.7267.432.5 3.774 2.89.958.733.443 3.7267.432.5 3.774 2.89.958 9.72 6.423.6892 4.83.7662 5.397 6.36 3.292 9.72 6.423.6892 4.83.7662 5.397 6.36 3.292 Table 8. Gains for the final design step, centralized control 2 of 8

2. 2..5.5.5.5 2 2 3 3. 2 3. 2 2 3 3. 2 3. 2.5 2.5.5.5 2 3.6. 2 3. 2 3.6. 2 3..4 V f V b.5 θ.4 V f V b.5 θ V.2 V.2.5.5.2 2 3. 2 3.2 2 3. 2 3 (a) decentralized coordinated control (b) centralized coordinated control Figure 7. Three Helicopters Coordinated Control of the Best Gains Possible: Responses to the Elevation Step Input...5.5 5 5.5.5. 2 3. 2 3. 2 3. 2 3.5.5 5 5.5.5. 2 3.4 2 3. 2 3.4 2 3.2 V f V b.2 V f V b V.2 θ V.2 θ.4 2 3 2 2 3.4 2 3 2 2 3 (a) decentralized coordinated control (b) centralized coordinated control Figure 8. Three Helicopters Coordinated Control of the Best Gains Possible: Responses to the Travel Step Input 3 of 8

model controller step e ε e ε2 e ε23 e λ e λ2 e λ23 E ε E λ state-space serial ε 774.6 575.9 55.3... 855.9. λ 36. 2825.6 2533.9 8959.5 decentr. ε 872.7 554.2 363.3... 79.. λ... 3685.9 3289.5 265.2. 94.6 centr. ε 842.6 29.2 9.4... 5.2. λ... 384.2 837.7 34.4 4993.2 nonlinear serial ε 776. 578. 58.3... 862.5. λ 6. 7.3 3.5 362.2 2828.7 2536. 6.7 8967. decentr. ε 874.6 556. 365.3... 796.. λ 5.4 6. 2.5 3687.4 329. 265.9 3.9 944.3 centr. ε 843.7 22.8 9.... 55.6..9 2.6.8 385.3 838.4 34.6 7.4 4995.2 HIL serial ε 83.7 6.8 537.9 53.2 97.8 53.8 969.4 24.7 2.3 9.7 5.3 3625.3 2947.5 269.5 47.4 9263.2 decentr. ε 949.9 592.7 386..5 4.8.8 928.7 327..3 9. 6.3 376. 3398. 23.6 46.7 9289.7 centr. ε 922. 252. 7.8 42.2 45.3 27.8 28.9 35.2 λ 9. 3.3 3.2 3889.8 948. 34.6 35.7 576.3 Table 9. Tracking error for initial design step model controller step e ε e ε2 e ε23 e λ e λ2 e λ23 E ε E λ state-space serial ε 774.6 575.9 55.3... 855.9. λ 36. 2825.6 2533.9 8959.5 decentr. ε 8.3 345. 35.... 46.5. λ... 4.7 232.2 528.4. 767.3 centr. ε 88. 64.6 6.7... 934.3. λ... 379.3 29.8 79.2 479.3 nonlinear serial ε 776. 578. 58.3... 862.5. λ 6. 7.3 3.5 362.2 2828.7 2536. 6.7 8967. decentr. ε 8.6 345.5 36.... 463...5 4. 2.7 4.5 233. 528.9.3 7673.5 centr. ε 89.3 65.3 62.3... 936.9....8 37.9 29.3 79.5 4.9 48.7 HIL serial ε 83.7 6.8 537.9 53.2 97.8 53.8 969.4 24.7 2.3 9.7 5.3 3625.3 2947.5 269.5 47.4 9263.2 decentr. ε 849.5 357. 338.9 52.2 67.7 4.5 545.3 224.5 λ 9.6 3.7 4.7 445. 285. 582.4 38. 782.4 centr. ε 85.7 9.4 88.3.8 74.9 76.6 3.4 252.3 λ 8.6 8.8.7 3847.3 37.5 79.9 29. 4334.6 Table. Tracking error for different configurations and inputs with best possible Q and R matrices 4 of 8

C. Three Helicopters: Hardware-In-The-Loop Experiments The results of the hardware-in-the-loop tests match the expectations from the simulations very well, as shown in Figures 9(a), 9(b), (a) and (b). Interestingly, the voltage needed to keep the helicopters hovering is around 2 V, whereas the nonlinear model needs less than.7 V. This means that the actual hardware model is heavier than assumed, but the controller compensates this modeling error well. Due to this difference, the elevation step response is a little bit slower than in the computer simulation. The travel step response is very close to the state-space performance. Figure (b) shows a peculiarity, observed throughout the whole testing period. Friction in the hinges prevents the helicopter to correct small errors in travel. The pitch angle increase until the static friction is overcome and the error is corrected. This behaviour only occurs for small deviations from the travel command. To complement the analysis where the step-input response was considered, a complete trajectory with several changes in elevation and travel was created. The setup consists again of three 3-DOF Helicopters. Figures and 2 show the elevation and travel as well as the errors between the helicopters and its commands for each of the three control strategies. Table shows the L 2 norms of the errors according to the equations in 4. For each strategy, the best possible setup was chosen. The tracking error for the first helicopter has the same magnitude for all strategies, but for the two followers, the differences in performance of the three controllers become apparent. For the elevation step, their tracking error is reduced by around 4 % from the serial setup to the decentralized controller and by more than 8 % for the centralized approach. For the travel step, the decentralized setup shows still better tracking performance than the serial setup, but the improvement is small. The centralized controller on the other hand exhibits extremely small tracking errors for the second and third helicopter. They are of a smaller order of magnitude than for the serial setup. The graph for travel of the serial setup shows that there is some overshoot, especially when a travel command is superposed with an elevation command. This can also be observed in the figures of the centralized controller. Since no overshoot was an important design criteria, the gain matrices should be modified in a further revision to compensate for this behaviour. But the extent of the overshoot is small and a modified gain matrix should not affect the other properties too much. VI. Conclusions This paper presents coordinated control among several laboratory three-degree-of-freedom (3-DOF) helicopters in their serial tracking maneuver. Both centralized and de-centralized design approaches are investigated using linear quadratic techniques. The numerical simulation and experimental hardware-in-the-loop results demonstrate the advantages and disadvantages of each approach. For further comparison, different tracking and synchronization configurations are currently under investigation. 5 of 8

2 2 2 2 2 2 3 3 2 3 3 2 2 3 3 2 3 2 2 2 2 2 3 4 2 3 2 3 4 2 2 3 V 3 V f V b 2 2 3 5 θ 5 2 3 V 3 2 V f V b 2 3 5 θ 5 2 3 (a) decentralized coordinated control (b) centralized coordinated control Figure 9. Three Helicopters Coordinated Control of the Best Gains Possible: HIL Responses to the Elevation Step Input.4.2.2 5.2 5.2.4.4 2 3.5 2 3.6 2 3.4 2 3 5.2 5.2.5 2 3 3 5 2 3.4 2 3 3 5 2 3 2.5 V f V b 2.5 V f V b V 2.5 θ V 2.5 θ 2 3 2 2 3 2 3 2 2 3 (a) decentralized coordinated control (b) centralized coordinated control Figure. Three Helicopters Coordinated Control of the Best Gains Possible: HIL Responses to the Travel Step Input model controller e ε e ε2 e ε23 e λ e λ2 e λ23 E ε E λ HIL serial 223 73 4 544 3869 342 3337 4254 decentr. 32 68 63 9479 694 8548 253 3972 centr. 275 63 76 64 34 86 64 863 Table. Tracking error for a complete trajectory scenario with different configurations and final gains 6 of 8

3 3 2 2 Elevation Travel 2 5 5 2 5 5 2 3 5 ε 3 2 5 5 5 5 5 2 5 5 Figure. HIL, decentralized control, tracking of a trajectory 3 4 2 3 Elevation Travel 2 2 5 5 2 5 5 2 3 5 2 5 5 5 5 5 2 5 5 Figure 2. HIL, centralized control, tracking of a trajectory 7 of 8

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